Influence of drive chamber discharging process on non-linear displacer dynamics and thermodynamic processes of a fluidic-driven Gifford-McMahon cryocooler

1 Introduction

Starting with the invention of the reverse – Stirling refrigerator by Philips Company of the Netherlands in 1950s and the Gifford-McMahon cycle (GMC) refrigerators in late 1960s by A.D. Little Inc., regenerative coolers have made an enormous impact on the growth of cryogenic technology around the world. Applications such as laboratory cryocoolers offering a 10 K environment for low-temperature experiments, refrigeration cryopumps, high T _c superconducting magnets, liquid helium re-liquefaction in MRI magnets, production of liquid nitrogen, liquid hydrogen and liquefied natural gas in laboratory-scale have contributed to wide acceptance of these devices, often called closed cycle cryocoolers or regenerative refrigerators [1, 2]. Commercial cryocoolers based on the reverse-Stirling and the GMCs are being sold in the international market for nearly half a century. The development of regenerative refrigerators has conquered technical barriers that were not even conceived three decades ago. The growing demand for high Tc superconducting materials in different areas of current science and technology desire an efficient cooling technology to retain its superconductivity state [1, 3]. Garceau et al. [4] adopted GM cryocoolers and heat pipe to make a robust small-scale hydrogen liquefier. Pneumatic-drive GMC is opted for making small scale hydrogen liquefiers of typical liquefaction rate of 1 L/h by Baik et al. [5]. Song et al. [6] proposed a novel carbon capture system using a Stirling cryocooler to achieve the cryogenic temperature. The GMC can be used in place of Stirling cryocooler because of its stable operation, higher MTBF and reliability. Paul et al. [7] developed a mathematical model that consists of a number of ordinary differential equations, and optimizes the piston paths of the cryocooler to maximize its efficiency. Ponce et al. [8] studied the non-linear processes of a thermoelectric cooler by mathematical modelling. In the other way, an enormous effort is given to analyse the displacer dynamics of GMC considering the influence of loss mechanisms like resistance loss, swing loss, void volume loss etc. Influence of all those loss mechanisms have been studied for a heat engine cycle by Li et al. [9]. Several numerical analyses [10–12], theoretical, thermodynamics and dynamic studies [13–16], and experimental investigations [17–19] are being carried out on GMC to magnify its cooling process.

Figure 1 sketches a schematic diagram of a fluidic-drive GMC to show its internal components. As shown in Figure 1, the drive chamber is coupled with the compressor supply line and compressor return line by means of secondary charging (Vsc) and discharging valves (Vsd). Drive chamber pressure magnitude depends on the opening-closing durations of Vsc and Vsd, i.e., the drive-chamber is filled with high and low pressure in the opening of Vsc and Vsd, respectively. Conversely, the pressure of compression and expansion chambers are controlled by primary charging valve (Vpc) and primary discharging valve (Vpd). The primary charging and discharging valves are together called as primary switching valve (Vps). The pressure inside the compression and expansion chamber is at high and low-pressure values during the opening of Vpc and Vpd, respectively. By opening the Vsc for a longer duration, pressure of drive chamber can be kept maximum for a longer duration and it will keep the displacer at bottom-dead-centre (BDC) for maximum durations in a cycle. Conversely, by opening Vsd for a longer duration, then pressure of drive chamber can be kept minimum value for a longer duration, and keeps the displacer at top-dead-centre (TDC) for maximum durations in a cycle. Therefore, the opening intervals of Vsc and Vsd control the displacer dynamics, volume changes of compression and expansion chambers, and ultimately the thermodynamics of the expansion chamber of fluidic-driven GMC. Therefore, it is essential to study the influence of the opening/closing duration of Vsc and Vsd on thermodynamic processes of the cryocooler, which has not been reported yet. Nevertheless, the valve-timing aspects of displacer cylinder valves of GMC are explored [14–16].

Figure 1:

Schematic diagram of fluidic-driven GMC.

A theoretical investigation is performed in present paper to examine the effect of secondary charging process/drive-chamber charging process and secondary discharging process/drive-chamber discharging process on the thermodynamic processes, displacer dynamics of a fluidic-driven GMC. Two terms are defined, such as drive-chamber charging process and drive-chamber discharging process (DCDP) to illustrate the opening durations of Vsc and Vsd, respectively. In the drive-chamber charging process, Vsc is opened, so the drive chamber is filled with high-pressure gas, and in DCDP, Vsd is opened, so the drive chamber is filled with low-pressure gas. Additionally, for different values of the DCDP, the transient flow rate distribution at the primary switching valve and regenerator are studied. Pressure changes of compression, expansion and drive chambers for different values of DCDP are elaborated. Additionally, pressure–volume (P–V) plots are drawn to show the path followed by gas parcels to generate the refrigeration effect for different durations of DCDP, and the relative deviation from ideal P–V plot is sketched for comparison purposes. Finally, its overall impact on the variation of P–V power in gas chambers, refrigerating capacity and Carnot efficiency is studied.

2 Mathematical background

The governing equations developed by Matsubara [20, 21] for a mechanically-driven GMC in combination with the displacer dynamics of fluidic-driven GMC of Minas et al. [22] are employed to estimate the thermodynamic processes of fluidic-driven GMC. Following Figure 1, the refrigerant of compressor is able to communicate with the displacer cylinder through primary charging valve and primary discharging valve during primary charging process and primary discharging process, respectively. Thus, flow rate through primary switching valve and regenerator can be calculated by employing Eqs. (1) and (2) [14].

On the other hand, pressure value of the drive chamber is taken in the code using Eqs. (3.a) and (3.b) for VT-II and VT-I, respectively.

Equations (4) and (5) determine the transient pressure change in expansion and compression chambers of fluidic-driven GMC and are derived using mass balance [14–16].

As mentioned below, the transient displacer position changes of the fluidic-driven displacer is modeled by utilizing SDOF with Coulomb damping [14, 22].

From Eq. (6), displacement (y) and velocity (z) of displacer oscillation can be calculated as follows:

The position change of compression and expansion chamber in each time of a cycle can be calculated as [16]:

Those four ODEs (i.e., Eqs. (1), (2), (7) and (8)) along with Eqs. (3)–(5), (9) and (10) estimates the flow rate through primary switching valve (primary charging valve plus primary discharging valve), regenerator, displacement and velocity of the displacer and pressure distribution in compression, drive and expansion chambers. These equations are integrated using Runge–Kutta method with an adaptive integration step. After the attainment of a cyclic steady state, the performance parameters (i.e., P–V power in expansion and compression chambers) are calculated using Eqs. (11) and (12), respectively [20, 21].

The results of the mathematical model are compared with Matsubara [20, 21] by omitting displacer dynamics of fluidic-driven displacer and using a sinusoidal volume change of expansion chamber. It is noticed that, the percentage deviation between present code with that of Matsubara [20, 21] is 1.96%. The same set of governing equations is employed to identify the thermodynamics of GMC, and the accuracy of the program with experimental values of an indigenous fluidic-driven GMC lies between 24 and 26% [14–16]. Therefore, it is opted to explore the influence of the drive-chamber discharging process on thermodynamic aspects of GMC.

2.1 Benchmark

Figure 2 illustrates two common types of valve-timing charts of a fluidic-driven GMC [14] and related terminology is written in Table 1. While investigating the influence of DCDP on the thermodynamic performance of the fluidic-driven GMC, the opening and closing intervals of Vpc and Vpd are fixed. The Vpc opens at 0° and closes at 175°, and Vpd opens at 180° and closes at 355°. The waiting period between Vsc and Vsd of the piston-cylinder is set to zero. The opening angle of Vsd is set to 40°, and its closing angle is computed depending upon the duration of DCDP in valve-timing I (VT-I), however, in valve-timing II (VT-II), the opening angle of the Vsd is set to 0°, and its closing angle is computed depending upon the duration of DCDP. Table 2 illustrates the opening-closing angles of the drive-chamber charging and discharging valves for different values of DCDP for both VT-I and VT-II, respectively. The stroke length of the displacer inside the displacer cylinder is 40 mm, and the diameter and length of the regenerator are 50 and 120 mm, respectively. The compressor discharge and suction pressure values are taken as 2.1 MPa and 0.76 MPa, respectively. The room temperature is set to 300 K, and refrigeration temperature is fixed to 80 K in the analysis [14–16].

Figure 2:

Valve-timing representations of both modes. (a) Valve-timing mode-I (VT-I) and (b) valve timing mode-II (VT-II). For description refer Table 1.

Table 1:

Valve-timing chart related terminology.

Serial number	Angle	Significance	Remark
1	α 1 (θ 1 to θ 2)	Primary charging period	Both VT-I and VT-II
2	α 2 (θ 2 to θ 3)	Waiting period	Both VT-I and VT-II
3	α 3 (θ 3 to θ 4)	Primary discharging period	Both VT-I and VT-II
4	α 4 (θ 4 to θ 1)	Waiting period	Both VT-I and VT-II
5	α 5 (θ 1 to θ 6 for VT-I) (θ 6 to θ 1 for VT-II)	Drive chamber charging process	Both VT-I and VT-II
6	α 6 (θ 6 to θ 7 for VT-I) (θ 1to θ 6 for VT-II)	Drive chamber discharging process (DCDP)	Both VT-I and VT-II
7	α 7 (θ 7 to θ 1)	Drive chamber charging process	Only for VT-I

Table 2:

Opening-closing angles of drive-chamber charging and discharging valves for different values of DCDP.

Duration of drive chamber	Opening intervals of α 6		Opening intervals of α 5		Opening intervals of α 7
discharging process (DCDP)	(VT-I)	(VT-II)	(VT-I)	(VT-II)	(VT-I)	(VT-II)
160°	40°–200°	0°–160°	0°–40°	160°–360°	200°–360°	–
180°	40°–220°	0°–180°	0°–40°	180°–360°	220°–360°	–
200°	40°–240°	0°–200°	0°–40°	200°–360°	240°–360°	–
220°	40°–260°	0°–220°	0°–40°	220°–360°	260°–360°	–
240°	40°–280°	0°–240°	0°–40°	240°–360°	280°–360°	–
260°	40°–300°	0°–260°	0°–40°	260°–360°	300°–360°	–
280°	40°–320°	0°–280°	0°–40°	280°–360°	320°–360°	–

3 Results and discussion

Figures 3 and 4 demonstrates the periodic fluctuation of displacer motion and expansion chamber volume for 160°, 200°, 220°, 260° and 280° of DCDP for both VT-I and VT-II, respectively. It is revealed that, during the primary charging process, high-pressure refrigerant enters to displacer cylinder, therefore pressure of compression/expansion chamber increases, as Vsd is open, the pressure of drive chamber is minimum value, and thus the high-pressure gas expels displacer to moves from the BDC to TDC. Alternatively, when Vsc opens, the drive-chamber charging process starts, so that pressure in drive chamber upsurges to its maximum value, and hence pushes the displacer to move from TDC to BDC. An increase in the duration of DCDP keeps the displacer at TDC for a longer period in a cycle, henceforth; the volume of expansion chamber is maximum for a longer period. It is noticed that, with an increase in the duration of DCDP, the expansion volume attains its maximum value for a longer interval in each working cycle for both VT-I and VT-II in Figure 4(a) and (b), respectively. As an unbalanced pressure is created between displacer cylinder gas chambers and piston-cylinder gas chamber at 40° of phase angle for VT-I, the displacer starts its motion at this angle, similar to reference [14]. But, it starts its movement at the starting of cycle in VT-II, because of the creation of unbalanced pressure values between displacer cylinder gas chambers and piston-cylinder gas chamber at the starting of cycle, in accordance with reference [16]. Additionally, it is identified that for a DCDP of 280° in VT-I, the displacer is in the BDC for a shorter interval in a cycle and at TDC for a longer interval in a cycle. Moreover, for DCDP of 280°, the expansion chamber volume is non-zero at 280°of phase angle, which signifies that the displacer is not located at the BDC, away from the BDC, and slowly moving towards BDC. Thus, for 280° of DCDP, volume variation of the fluidic-driven GMC is like an asymmetric GMC [23, 24], and its thermodynamic cycles are reported elsewhere [15]. However, in VT-II an asymmetric motion of displacer oscillation has not been detected in this work, and its thermodynamic processes are elaboared elsewhere [16].

Figure 3:

(a) Influence of drive chamber discharging process on displacement of displacer according to VT-I. (b) Influence of drive chamber discharging process on displacement of displacer according to VT-II.

Figure 4:

(a) Influence of drive chamber discharging process on volume variation of expansion chamber according to VT-I. (b) Influence of drive chamber discharging process on volume variation of expansion chamber according to VT-II.

Figure 5(a) and (b) illustrates the synchronous influence of DCDP and waiting time on cycle average flow rate at primary charging valve for both VT-I and VT-II, respectively. It has been seen that the cycle average flow rate reduces with an increase in waiting time and DCDP because of closing durations of primary and secondary valves.

Figure 5:

(a) Influence of drive chamber discharging process on cycle average flow rate according to VT-I. (b) Influence of drive chamber discharging process on cycle average flow rate according to VT-II. Mass flow rate unit is in [g/s].

The pressure at gas chambers of displacer cylinder and piston-cylinder, as well as expansion chamber volume have been plotted in Figures 6 and 7 for VT-I and VT-II, respectively, for 160°, 200°, 220°, 240°, 260° and 280° of DCDP. During the primary charging process, displacer cylinder is in connection with the discharge port of compressor, thus, pressure of compression and expansion chambers increases higher to that of mean pressure, and the opposite case happens in the next half-cycle. This difference in the pressure and volume in primary charging and discharging processes because of different DCDP tends to create different shapes of P–V curves in the expansion chamber. It is also noticed that, due to an increase in the duration of DCDP, the pressure in the drive chamber remains a low-pressure value for a longer period. For DCDP of 160°, 200°, 220°, 240°, 260° and 280°, the drive chamber is 0.76 MPa between 40° and 200°, 40°–240°, 40°–260°, 40°–280°, 40°–300° and 40°–320°, respectively, for VT-I. On the other hand, for DCDP of 160°, 200°, 220°, 240°, 260° and 280°, the drive chamber is 0.76 MPa between 0° and 160°, 0°–200°, 0°–220°, 0°–240°, 0°–260° and 0°–280°, respectively, for VT-II. During a pressure change in the drive chamber from the high-pressure to the low-pressure value, the displacer starts its movement from BDC to TDC, and vice-versa. In these angles, (i.e., displacer moves from BDC to TDC, or from TDC to BDC as noticed from Figure 3(a) and (b) for VT-I and VT-II, respectively), a clear and distinguishable pressure difference exists between expansion and compression chambers for both VT-I and VT-II as noticed in Figures 6 and 7. Accordingly, the position traced by the displacer changes by a marginal value, and the expansion chamber volume path is varied as explained earlier in Figure 4(a) and (b).

Figure 6:

Pressure of expansion chamber, compression chamber, drive chamber and volume of expansion chamber for different values of drive-chamber discharging process according to VT-I.

Figure 7:

Pressure of expansion chamber, compression chamber, drive chamber and volume of expansion chamber for different values of drive-chamber discharging process according to VT-II.

Figure 8 demonstrates a comparable illustration of the ideal P–V curve and theoretical/real P–V curve of the expansion chamber for 160°, 200°, 220°, 240°, 260° and 280° of drive-chamber discharging process for VT-I. As suggested by Hao et al. [17] and Matsubara [20], the ideal P–V cycle gets completed in four processes. However, a small slated cut-off is present in the bottom-left and top-right corners of the ideal P–V loop because of 5° waiting periods of primary switching valves as explained in reference [14]. However, the real P–V diagram in Figure 9 uses six thermodynamic processes to demonstrate a complete cycle [14]. The expansion chamber volume is minimum and pressure increases from state “i” to state “ii” because of charging of refrigerant by primary charging valve in process “i” to “ii”. Displacer travels from BDC to TDC, and therefore, expansion chamber volume increases from lower to higher limits, and pressure is around its higher value during the process “ii” to “iii”. Although the displacer reaches the TDC, pressure in the expansion chamber increases from “iii” to “iv” due to the continuous charging of refrigerant from discharge port of compressor by primary charging valve. In process “iv” to “v”, the displacer is at TDC, but pressure drops because of the opening of primary discharging valve and a cooling effect is generated because of expansion of refrigerant in expansion chamber. From “v” to “vi”, the displacer moves from TDC to BDC, and pressure continuously drops in expansion chamber because of unseating of primary discharging valve. At “vi”, the displacer reached the BDC, but pressure drops due to the opening of primary discharging valve. Therefore, six processes with six points are sketched by the gas parcel to complete one cycle, as elaborated in reference [14]. The parameters of each point are tabulated in Table 3 for VT-I. The shading areas in Figure 8 show the power loss due to different drive-chamber discharging process. Figure 9 depicts a clear view of the real/theoretical P–V curve at the expansion chamber for various DCDPs, such as 160°, 200°, 220°, 240°, 260° and 280° for VT-I without the ideal paths. The P–V diagrams follow six processes as discussed earlier. But, due to the change in state variables of pressure and volume due to different DCDP as explained in Figure 6, the nature of the P–V diagram changes. Also, for a DCDP of 260°, and 280°, point “i” and “vi”, coincides with each other. Therefore, the thermodynamic cycle gets completed in five processes. This is due to the recompression of the refrigerant during the primary discharging process before the starting of the primary charging process and volume change of the expansion chamber. Therefore, displacer motion inside displacer cylinder can be controlled and its locus is approximately shown by an asymmetric pneumatically-driven GM cryocooler, and its thermodynamics cycles is explained in reference [15].

Figure 8:

Ideal and real P–V diagram in the expansion chamber for different values of drive chamber discharging processes according to VT-I.

Figure 9:

Real P–V diagram in the expansion chamber for different values of drive chamber discharging processes according to VT-I.

Figure 10 demonstrates the ideal and real/theoretical P–V plots at the expansion chamber of fluidic-driven GMC for 160°, 200°, 220°, 240°, 260° and 280° of drive-chamber discharging process for VT-II. Here, also by employing four thermodynamic processes, the ideal gas cycle is demonstrated, and by utilizing six processes, the theoretical model gas cycle is demonstrated. Essential parameters of each point are tabulated in Table 4 for VT-II. The physical significance of each thermodynamic process is similar to reference for an ideal cycle, and the theoretical/real model gas cycle is similar to that of VT-I as explained earlier. A clear view of theoretical/real P–V plot is redrawn in Figure 11 to elaborate the thermodynamic processes of gas cycle. It is revealed that for 260° and 280° of drive-chamber discharging process in VT-II, a cycle required six processes, but five processes are needed for VT-I in these DCDP to complete a cycle.

Figure 10:

Ideal and real P–V diagram in the expansion chamber for different values of drive chamber discharging processes according to VT-II.

Figure 11:

Real P–V diagram in the expansion chamber for different values of drive chamber discharging processes according to VT-II.

Figure 12(a) and (b) illustrates the simultaneous effects of DCDP and waiting time on the P–V power of the expansion chamber for VT-I and VT-II, respectively. Similarly, Figure 13(a) and (b) illustrates the influences of DCDP and waiting time on the P–V power of the compression chamber of GMC for VT-I and VT-II, respectively. The refrigeration capacity is estimated after subtracting the P–V-power loss because of inherent loss mechanisms, and its variation versus DCDP and waiting time are elaborated in Figure 14(a) and (b) for VT-I and VT-II, respectively. It is seen that, with an increase in the duration of DCDP, the P–V power in the expansion chamber, compression chamber, and refrigerating capacity increases, becoming maximum value at 240° of DCDP for VT-I and 260° of DCDP for VT-II, and then drops. This is because the enclosed area is maximum under the P–V diagram for 240° of DCDP for VT-I and 260° of DCDP for VT-II, as noticed from Figures 9 and 11, respectively (waiting time = 5° in Figures 9 and 11). However, with an increase in waiting time, the P–V power reduces for VT-I and VT-II, which agrees with earlier results [14, 16].

Figure 12:

(a) Influence of drive chamber discharging process and waiting time on P–V power of expansion chamber according to VT-I. (b): Influence of drive chamber discharging process and waiting time on P–V power of expansion chamber according to VT-II.

Figure 13:

(a) Influence of drive chamber discharging process and waiting time on P–V power of compression chamber according to VT-I. (b) Influence of drive chamber discharging process and waiting time on P–V power of compression chamber according to VT-II.

Figure 14:

(a) Influence of drive chamber discharging process and waiting time on refrigeration capacity according to VT-I. (b) Influence of drive chamber discharging process and waiting time on refrigeration capacity according to VT-II.

Figures 15 and 16 shows the P–V-power loss at the primary charging valve section and primary discharging valve section for both types of valve-timing arrangements of the cryocooler, respectively. It is seen that the valve loss at the primary charging valve and the primary discharging valve reduces for both types of valve-timing arrangements with an increase in waiting time and DCDP. This is due to the decrease in flow rate with the rise in the waiting period. But, DCDP is the more dominating factor in P–V power degradation over the waiting time as seen from the graphs. Therefore, a careful selection of the DCDP will give a maximum cooling performance from the cryocooler. The coefficient of performance measures the ratio of refrigerating capacity per input compressor work. The variation of percentage Carnot efficiency against DCDP and waiting time are illustrated in Figure 17(a) and (b) for VT-I and VT-II, respectively. It has been noticed that maximum Carnot efficiency can be attained at a DCDP of 240° and 260° for VT-I and VT-II, respectively.

Figure 15:

(a) Influence of drive chamber discharging process and waiting time on valve loss at primary charging valve according to VT-I. (b) Influence of drive chamber discharging process and waiting time on valve loss at primary charging valve according to VT-II.

Figure 16:

(a) Influence of drive chamber discharging process and waiting time on valve loss at primary discharging valve according to VT-I. (b) Influence of drive chamber discharging process and waiting time on valve loss at primary discharging valve according to VT-II.

Figure 17:

(a) Influence of drive chamber discharging process and waiting time on percentage of Carnot efficiency according to VT-I. (b) Influence of drive chamber discharging process and waiting time on percentage of Carnot efficiency according to VT-II.

An experimental investigation is conducted for validation of result on the existing test-rig [14–16]. The schematic diagram of the experimental test rig with positions of measuring instruments is shown in Figure 18, and laboratory view is shown in Figure 19. We have used calibrated Pt-100 Ω temperature sensors and heater to measure the temperature variation and cooling load [14–16]. The type A and type B uncertainty calculation processes of measuring instruments are elaborated elsewhere [14–16]. Structural parameters are similar to the numerical values, and similar to the earlier data except the valve block [14–16]. Figure 20(a) and (b) shows that a numerical and experimental cooling capacity along with the associated uncertainty for VT-I and VT-II, respectively. The numerically computed cooling capacity and experimentally observed cooling capacity consistently increase with an increase in temperature for both VT-I and VT-II, respectively. A good agreement is noticed between numerical and experimental cooling capacity, and the small percentage of deviation is because of the real gas loss, fabrication loss, undetected leakage loss and contact resistance measurement loss.

Figure 18:

Schematics of experimental test rig.

Figure 19:

Laboratory view of experimental test rig.

Figure 20:

(a) Experimental and numerical capacity map according to VT-I. (b) Experimental and numerical capacity map according to VT-II.

The simulation model also computes the influence of the DCDP upon the various loss quantities and is tabulated in Tables 5 and 6 for 160°, 200°, 240° and 280° of DCDP for both VT-I, and VT-II, respectively, for fixed value of waiting time (5°). It is observed that at 260° and 280° of DCDP, the maximum percentage of supplied acoustic work is converted into the cooling load for VT-I and VT-II. Table 7 compares the COP of GMC in different refrigeration temperatures with the COP of the ideal GM C, Carnot cycle, Brayton cycle and Brayton cycle with no work recovery [16]. As the pressure ratio is approximately 2.763, the COP of the ideal GM cycle is comparatively lower than the COP of the Brayton cycle with and without work recovery at all refrigeration temperatures [16]. Within the ranges of refrigeration temperature, the COP of GMC for different DCDP is lower than the COP of the ideal GM cycle. Additionally, the COP of GMC for VT-II is lower than the COP of GMC with VT-I. Thus, choosing an accurate valve-timing arrangement is essential to achieve optimum performance from a fluidic-driven GMC.

4 Conclusions

In this paper, a mathematical model is employed to calculate the non-linear thermodynamic processes, dynamic behaviors, and fluid flow aspects of fluidic-driven GMC for different drive-chamber discharging processes. This mathematical model is computationally faster than that of the nodal analysis models. By using the model, effect of drive-chamber discharging process on the thermodynamic performance of GMC is evaluated. The following conclusions are obtained:

1. It is revealed that, by increasing the duration of the drive-chamber discharging process, it is possible to keep the displacer at its upper dead center for a longer duration in a cycle in valve-timing I. Thus, the motion of the displacer can be controlled. The displacer is allowed to move faster in the bottom dead center and allowed to move slower near the top dead center of the displacer cylinder. In this type of arrangement, the expansion chamber volume sketch is an “asymmetric pneumatic drive GM cycle”. In an asymmetric fluidic-driven GM cycle, the flow rate at the regenerator is non-zero because of displacer motion at the start of the cycle. But, valve-timing arrangement II, it is not possible to get an asymmetric GMC.

2. The cycle average mass flow rate at the primary charging valve decreases with an increase in the drive chamber discharging process for both types of valve-timing arrangements.

3. The P–V power of the compression chamber, P–V power of the expansion chamber, and refrigerating capacity increase with an increase in the duration of the drive-chamber discharging process, becomes maximum at 240° of the drive-chamber discharging process, and then decreases for the first valve-timing arrangement.

4. The P–V power of the compression chamber, P–V power of the expansion chamber, and refrigerating capacity increase with an increase in the duration of the drive-chamber discharging process, becoming maximum at 260° of drive-chamber discharging process, and then decreases for the second valve-timing arrangement.

5. The maximum percentage of supplied work is converted into useful cooling power approximately at 260° and 280° of the drive-chamber discharging process for both first and second valve-timing arrangements.

6. P–V power of compression chamber, P–V power of expansion chamber and refrigeration capacity of fluidic-driven GMC is relatively higher for first valve-timing arrangement than that of the second valve timing arrangement.

7. The ideal COP of GM cycle is found to be lower than the COP of Brayton cycle with and without work recovery, and Carnot cycle for these conditions due to moderately higher-pressure ratio. At each refrigeration temperature, the COP of second valve-timing is lower than that of the COP of first valve-timing.