Effects of Compressor Frequency on Performance of Inertance Tube Pulse Tube Refrigerator: A Numerical Study

. Pulse-tube refrigerators are considered as the mainstream elements in cryogenic plants. Normally, the efficacy of the inertance tube pulse tube refrigerator (ITPTR) is considered to be the highest among other pulse-tube refrigerators. In the current study, the mechanical performance of ITPTR system is numerically investigated to identify the impact of compressor frequency. The finite volume approach (FVM) is utilized to model the whole ITPTR with the specified boundary conditions using a commercial program ANSYS. The modelled ITPTR includes an inertance tube, reservoir, pulse-tube, cold heat exchangers, after cooler, regenerator, compressor and hot heat exchangers. These coupled systems that are simulated in a steady-state to reveal the performance and cooling rate. It has been discovered that the compressor's frequency has a significant impact on the ITPTR's performance and cooling rate. In four distinct ways, the compressor's frequency has been changed from 10 Hz to 100 Hz. When the compressor runs at 20 Hz, the rate of cooling is determined to be at its highest.


Introduction
Pulse tube refrigerators (PTR) are porous tubes with cooling properties on one end and pulsating pressure on the opposite end.It falls into a category of cryo-coolers that can attain cryogenic temperatures at low temperatures without a moving element.Basic pulse-tube refrigerators (BPTR) are the truly advanced category of cryogenic cooling system.In cryogenic refrigeration, the pulse tube is crucial because there are no movable elements in the low-temperature zone.The lack of moving components in this device makes it more reliable and durable than the Stirling and Gifford-McMahon system, especially in cold conditions.PTRs are more reliable than current cryo-coolers because of their simple design.Typically, these devices work well for cooling at temperatures lower than 120 K.The study on pulse tube refrigerator applications has recently received more interest.Numerous researchers have expressed interest in the PTRs over the years.These PTRs are extensively studied over the past several decades.The pressurised laboratory scale PTR was investigated by Moldenhauer et al. [1] utilising experimental design methods.The attributes of the OPTR system is investigated for a range of working frequencies with helium as the working fluid, orifice diameters, and mean pressures (3.5 bar to 22 bar) [2].Furthermore, a 2D computational fluid dynamics (CFD) analysis on a GMtype OPTR system was carried out by Zhang et al. [3].Likewise, a stirling-type U-shape PT based cryo-cooler with two-stages was modelled and fabricated by Yang et al. [4].This system was governed by a linear type compressor with maximum capacity of 10 kW.Farouk et al. [5] investigated the flow and operation of a single phase coaxial OPTR.Moreover, the performance of the OPTR was enhanced by reducing the friction generated by the flow of gas, regenerator, heat transfer in the heat exchanger and material properties [6].Similarly, the effects of multiphase working fluids consisting of helium and nitrogen mixtures as the ingredients were studied for a single phase PTR system [7].It worked better because they were able to get the same temperature as with the model, and it only used working fluid as helium.The refinement in the efficiency of a single-stage PTR, Gao et al. [8] contrasted previous PTR gas mixtures.Based on the findings of the preliminary research, Liang et al. [9] developed the complex PTR system.It provided the coupled thermo-viscous effects of the wall, and made a prediction about the presence of a thermo-interfacial layer inside the PTR.From the experimental analysis at CEA/SBT on PTR, it was confirmed that the features of the steadyflow had significant effect on the mechanical performance in the dual inlet system [10].In order to build a BPTR model that creates a temperature profile that is more accurate, De Boer et al. [11] modified a number of the model's components.One of these was gas motion, which included both cooling and heating effects.They also examined the BPTR with heat exchangers and regenerators located at opposite ends.
It is discovered from the thorough literature review that various researchers have tried to do experimental study on PTR.But there are still certain instances where there aren't any conclusive results, urging for a complete analysis of PTR's efficiency.The goal of this study is to quantitatively examine the ITPTR performance when the compressor frequency is modified while using nitrogen as the working fluid.

Finite volume methodology (FVM)
The FVM approach is employed in this research as it is capable of providing accurate solutions by employing an efficient approach.Thus, this technique is certainly reliable for obtaining the governing equations for complex problems involving fluid dynamics, heat and mass transfer problems.Meanwhile, the FVM's ability to satisfy parameters such as energy, mass, and momentum is one of its most intriguing characteristics.A result with a coarse grid displays precise integral balances.In addition, it can be applied to extremely intricate geometries and any grid (structured or unstructured, cartesian, coarse or fine).As a consequence, the majority of commercial products, such as ANSYS, CFX and CHX are employed to resolve problems involving fluid dynamics, heat and mass transfer problems adhere to this strategy.In this FVM technique, it divides the problem domain to continuous compartments or control volumes (CV).The input variables are positioned at the centroid of the CV.Thus, the grids are established according to it.Integration of the differential governing equations is the further step in each control volume.Furthermore, interpolation process is performed using various techniques like central differencing (CD), upwind differencing (UD), power-law differencing (PD), and quadratic upwind differencing (QUD).Therefore, the FVM technique utilized the discrete equations to solve the complex problems.This discretized equation expresses the hypothesis of variable conservation within the CV.These input factors constitute a group of simultaneous algebraic computations that can be solved using a particular algorithm.
In this investigation, the commercial ANSYS user-defined code (UDF) is utilised.It is expected that the simulated ITPTR systems have a cylindrical and linear alignment.Axisymmetric and two-dimensional flows are also considered.The basic UDF for piston motion can be represented as: Where = 0.004511m, = 213.62rad/sec and the time increment of 0.00073529 s.ANSYS is responsible for solving the equations relating to mass (m), momentum (p), and energy (E) for the simulation.These equations are described as: Where Nitrogen serves as the "working fluid".The equations shown above may be applied to every component of the ITPTR, with the exception of the cold heat exchangers, after cooler, regenerator, compressor and hot heat exchangers.These four parts can be modeled as porous media by using the following mass, momentum & energy equations by taking = 0.69,

Model consideration for simulation
Fig. 1 illustrates the configuration of the ITPTR used for the simulation.For the gas within the system, compressor (A) generates a harmonic oscillation.The compressor is connected to the after-cooler component via transfer line (B).During the compression process, an after cooler (C) is used to remove heat from the compressor.Regenerator (D) is a porous matrix that can exchange heat with gas, and it has the superior properties like retention of excessive heat with extreme low thermal conductivity.Through the compressor's forward and backward strokes, heat from the incoming gas is absorbed and then returned to the gas.During the cooling phase in the expansion cycle, the cold heat exchanger (E) takes the heat from surrounding environment.By performing refrigeration at the input end and pulsating pressure at the output end, a pulse tube (F) is used to pump the heat out of a cold heat exchanger.Through the heat exchanger (G), the generated heat during the compression cycle is released to the sourrounding.The reservoir is coupled to the heated heat exchanger via the inertance conduit (H).It is a very prolonged intake for flow resistance in terms of size and diameter.The surge volume (I) is significantly larger than the remainder of the system, and gas is beginning to migrate back from the reservoir wall, producing a cooling effect.Table 1 presents the geometric dimensions of all the components.These dimensions remain unchanged for frequencies like 10 Hz to 100 Hz.In order to facilitate UDF for compressors with variable frequencies, the numerical analysis of PTR is aided by dynamic meshing.
There was piston movement within the compressor.For the simulation, the boundary conditions provided for the numerous ITPTR components are detailed in Table 2.A magnified image of the axis-symmetric geometries with meshing is shown in Fig. 2.  , 01

Results and Discussion
This section includes the details of the external temperature of a cold heat exchanger under no-load conditions.It is considered during the discussion of CFD simulation results.For each frequency condition, distinct graphs depict cooling behaviour and cold heat-exchanger temperature.In addition, the contour diagram for steady-state surface temperature change is investigated.

ITPTR results at frequency 10 Hz
The simulation findings in Fig. 3 show that ITPTR keeps the cold heat exchanger temperature constant under noload situations.Here, the piston of the compressor is moving at a rate of 10 Hz.With a frequency of 10 Hz, the cooling action and surface temperature distribution of the cold-end heat exchanger are depicted below.In this instance, the temperature of the cold end of the heat exchanger is 277.62 K.

ITPTR results at frequency 20 Hz
The simulation findings in Fig. 4 suggest that ITPTR maintains the cold heat exchanger temperature under no-load environments.Here, the piston of the compressor is moving at a rate of 20 Hz.With a frequency of 20 Hz, the cooling characteristics and surface temperature distribution of the cold-end heat exchanger are depicted below.In this instance, the temperature of the cold end of the heat exchanger is 264.91 K.

ITPTR results at frequency 60 Hz
The results of this simulation are shown in Fig. 5, and they demonstrate that ITPTR is capable of preserving the temperature of the cold heat exchanger even when there is no load present.The piston of the compressor is running at 60 Hz in this case.The frequency of 60 Hz is used to illustrate the cooling behavior and surface-level temperature distribution of the cold-end heat exchanger.In this particular instance, the temperature at the cold-end of the heat exchanger is determined to be 270.49K.

ITPTR results at frequency 100 Hz
The modelling results in Fig. 6 suggest that when there is no load, ITPTR keeps the cold heat exchanger at the same temperature.In this case, the piston of the expander is moving at a rate of 100 Hz.With a frequency of 100 Hz, the cooling action and surface temperature distribution of the cold-end heat exchanger are shown below.In this instance, the temperature of the cold-end of the heat exchanger is 274.21K.

Comparison of ITPTR results for various frequencies
The following graph (Fig. 7) demonstrates the comparison of the cooling effects of the four frequency investigations.In this particular instance, the comparison is accomplished by sketching a graph in which time (expressed in seconds) is plotted along the X-axis, and temperature (expressed in Kelvin) is shown along the Y-axis.It has been determined that the rate of cooling at 20 Hz for the aforementioned dimension is better than that at other frequencies.

Conclusions
The ITPTR systems were modelled numerically using the ANSYS software while operating in a stable periodic state with the same boundary conditions and varying frequencies.The primary objectives were to demonstrate the viability of PTR CFD modelling and examine the multidimensional flow as well as heat transfer effects.CFD models accurately predicted all expected trends.It is discovered that the frequency of the compressor has a significant effect on the efficacy and cooling rate of the ITPTR.The frequency range of the compressor is 10 Hz to 100 Hz.At a compressor frequency of 20 Hz, the rate of cooling is determined to be the greatest.

Fig. 2 .
Fig. 2. Magnified meshed image of the axis-symmetric model of various parts.

Fig. 7 .
Fig. 7. Comparison of cooling effects of cold-end heat exchanger at different frequencies.

Table 2 .
Numerical boundary conditions for various ITPTR components.